What are the solutions to the equation (x-1)^7=x^5 ?

The solutions to the equation (x-1)^7=x^5 are x\approx 3.39350…

Find the zeros of (x-1)^7=x^5

The zeros of (x-1)^7=x^5 are x\approx 3.39350…

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(x - 1)^{7} = x^{5}

x \approx 3.39350 \dots

Solution steps

(x - 1)^{7} = x^{5}

Expand

(x - 1)^{7} : x^{7} - 7 x^{6} + 21 x^{5} - 35 x^{4} + 35 x^{3} - 21 x^{2} + 7 x - 1

x^{7} - 7 x^{6} + 21 x^{5} - 35 x^{4} + 35 x^{3} - 21 x^{2} + 7 x - 1 = x^{5}

Move

x^{5}

to the left side

x^{7} - 7 x^{6} + 20 x^{5} - 35 x^{4} + 35 x^{3} - 21 x^{2} + 7 x - 1 = 0

Find one solution for

x^{7} - 7 x^{6} + 20 x^{5} - 35 x^{4} + 35 x^{3} - 21 x^{2} + 7 x - 1 = 0

using Newton-Raphson:

x \approx 3.39350 \dots

Apply long division:

\frac{x ^{7} - 7 x ^{6} + 20 x ^{5} - 35 x ^{4} + 35 x ^{3} - 21 x ^{2} + 7 x - 1}{x - 3.39350 \dots} = x^{6} - 3.60649 \dots x^{5} + 7.76134 \dots x^{4} - 8.66182 \dots x^{3} + 5.60601 \dots x^{2} - 1.97592 \dots x + 0.29468 \dots

x^{6} - 3.60649 \dots x^{5} + 7.76134 \dots x^{4} - 8.66182 \dots x^{3} + 5.60601 \dots x^{2} - 1.97592 \dots x + 0.29468 \dots \approx 0

Find one solution for

x^{6} - 3.60649 \dots x^{5} + 7.76134 \dots x^{4} - 8.66182 \dots x^{3} + 5.60601 \dots x^{2} - 1.97592 \dots x + 0.29468 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solution is

x \approx 3.39350 \dots

12 x^{3} + 21 x^{2} + 6 x = 0

x^{3} - 7 x^{2} + 17 x - 15 = 0

x^{4} + x^{2} = 2

x^{8} = 3^{16}

- x^{3} - x^{2} + 4 x + 4 = 0

What are the solutions to the equation (x-1)^7=x^5 ?
The solutions to the equation (x-1)^7=x^5 are x\approx 3.39350…
Find the zeros of (x-1)^7=x^5
The zeros of (x-1)^7=x^5 are x\approx 3.39350…